Theorem denexp

Description: Elevating to a nonnegative power commutes with canonical denominator. Similar to densq , extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023)

	Ref	Expression
Assertion	denexp	⊢ ( ( 𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0 ) → ( denom ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( denom ‘ 𝐴 ) ↑ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		numdenexp	⊢ ( ( 𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0 ) → ( ( numer ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( numer ‘ 𝐴 ) ↑ 𝑁 ) ∧ ( denom ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( denom ‘ 𝐴 ) ↑ 𝑁 ) ) )
2	1	simprd	⊢ ( ( 𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0 ) → ( denom ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( denom ‘ 𝐴 ) ↑ 𝑁 ) )